Calculus III
Practice Problems 2
1. Find a unit vector orthogonal to the vectors V 6I 7J
the parallelopiped determined by these three vectors?
K W
I
2J 3K. What is the volume of
2. Find a vector which makes an angle of 30 with the plane determined by the vectors V
I 2J 3K.
6I 7J K W
3. Find a vector normal to the plane given by the equation
3x 2y
z
14 4. Find a vector normal to the plane through P : 0 0 0 Q : 1 0 1
R : 0 1 1 5. Find the equation of the plane through the origin which is normal to the line given parametrically by
3I
X
2J K
t I
J
2K
6. Find V1 V2 V3 where
V1
I
2J
K
2I 2J
V2
3K V3
I 2K 7. Show that if det U V W U V W , then the three vectors are mutually orthogonal.
8. Find the symmetric equations of the line through the point (2,-1,3) which is perpendicular to the vectors
V 2I J 3K and W I J K.
9. Find the parametric equations of the line through the point (2,-1,3) which is parallel to the two planes given
by the equations
3x z 4 x 2y 5z 1 10. Find the equation of the plane through the point (2,-1,3) which is parallel to the vectors I 2J
3I 2J K.
3K and
11. Find the distance of the point 2 0 1 from the plane given by the equation
x 2 y 1 z 1
3
4
2
0
12. Let L1 L2 be two lines in three dimensions which do not intersect. There are points P1 P2 on L1 L2
respectively such that the line joining P1 and P2 intersects each line at a right angle. These are the points on
the two lines which are closest together. Find a formula for the distance between P1 and P2 , in terms of the
equations of the lines
L2 : X Q2 sN2 L1 : X Q1 tN1